Introduction
dynamical behavior along the center direction
Main results
Proof of Main results:
Dynamical behavior of ergodic measure along the
weak direction
Second Palis-Balzan International Symposium on Dynamical
Systems
Jiagang Yang
UFF
Departamento de Geometria
[email protected]
June 2013
Jiagang Yang
Dynamical behavior of ergodic measure along the weak directio
Introduction
dynamical behavior along the center direction
Main results
Proof of Main results:
Partially hyperbolic diffeomorphisms
A diffeomorphism f is partially hyperbolic if the tangent space
splits into E s ⊕ E c ⊕ E u . We say f is dynamically coherent if it
admits an invariant center foliation.
Jiagang Yang
Dynamical behavior of ergodic measure along the weak directio
Introduction
dynamical behavior along the center direction
Main results
Proof of Main results:
Partially hyperbolic diffeomorphisms
A diffeomorphism f is partially hyperbolic if the tangent space
splits into E s ⊕ E c ⊕ E u . We say f is dynamically coherent if it
admits an invariant center foliation.
3-dimensional examples:
• (Flow type): f is the perturbation of the time-one map of an
Anosov flow.
Jiagang Yang
Dynamical behavior of ergodic measure along the weak directio
Introduction
dynamical behavior along the center direction
Main results
Proof of Main results:
Partially hyperbolic diffeomorphisms
A diffeomorphism f is partially hyperbolic if the tangent space
splits into E s ⊕ E c ⊕ E u . We say f is dynamically coherent if it
admits an invariant center foliation.
3-dimensional examples:
• (Flow type): f is the perturbation of the time-one map of an
Anosov flow.
• (Skew product type) f is the perturbation of a skew product
map over T 2 × S 1 : f ((x, θ)) = (g (x), hx (θ)).
Jiagang Yang
Dynamical behavior of ergodic measure along the weak directio
Introduction
dynamical behavior along the center direction
Main results
Proof of Main results:
Partially hyperbolic diffeomorphisms
A diffeomorphism f is partially hyperbolic if the tangent space
splits into E s ⊕ E c ⊕ E u . We say f is dynamically coherent if it
admits an invariant center foliation.
3-dimensional examples:
• (Flow type): f is the perturbation of the time-one map of an
Anosov flow.
• (Skew product type) f is the perturbation of a skew product
map over T 2 × S 1 : f ((x, θ)) = (g (x), hx (θ)).
• (Derived from Anosov type) f ∈ Diff 1 (T 3 ) is in the isotopic
class of a linear Anosov diffeomorphism A0 , which has three
real eigenvalues with different norms.
Jiagang Yang
Dynamical behavior of ergodic measure along the weak directio
Introduction
dynamical behavior along the center direction
Main results
Proof of Main results:
Dynamical behavior along the center direction
The dynamics of volume in the center direction are more
complicated than in the strong direction. There are the following
three cases:
(1) (Fubini’s nightmare): there is a full volume subset, which
intersects each center leaf with finitely many points.
(2) (Weak absolute continuation): The disintegration of volume
along the center direction is absolutely continuous respect, but
not equivalent to the Lebesgue measure along the center leaf.
(3) (singular but not atomic case) The disintegration of volume
along the center direction is not atomic, and has no any
absolutely continuous property.
Jiagang Yang
Dynamical behavior of ergodic measure along the weak directio
Introduction
dynamical behavior along the center direction
Main results
Proof of Main results:
Two examples for Fubini’s nightmare
Example 1 (Katok):
f : S 1 × I → S 1 × I : f (x, t) = (ht (x), t) where
• ht is uniformly expanding;
• ht (0) = 0 for any t ∈ I and Dhs (0) 6= Dht (0) for any
s 6= t ∈ I .
Jiagang Yang
Dynamical behavior of ergodic measure along the weak directio
Introduction
dynamical behavior along the center direction
Main results
Proof of Main results:
Two examples for Fubini’s nightmare
Example 1 (Katok):
f : S 1 × I → S 1 × I : f (x, t) = (ht (x), t) where
• ht is uniformly expanding;
• ht (0) = 0 for any t ∈ I and Dhs (0) 6= Dht (0) for any
s 6= t ∈ I .
Example 2 (Ruelle, Wilkinson):
Let f be a C 2 skew type of partially hyperbolic, volume preserving
diffeomorphism. Assume that f is ergodic, and λc (f ) > 0.
See also the results of Avila, Viana and Wilkinson in the case the
center exponent is vanishing.
Jiagang Yang
Dynamical behavior of ergodic measure along the weak directio
Introduction
dynamical behavior along the center direction
Main results
Proof of Main results:
Fubini’s nightmare
Fix a diffeomorphism f of skew type or flow type. For α > 0,
denote
Γα (f ) = {x ∈ M : lim inf
1
log |Df n |E c (x) | > α.}
n
Proposition (Viana, Y)
There is δ such that
x ∈ Γα (f ) and every neighborhood
P for every
U of x, lim inf n1
length(f i (U)) > δ.
Jiagang Yang
Dynamical behavior of ergodic measure along the weak directio
Introduction
dynamical behavior along the center direction
Main results
Proof of Main results:
Fubini’s nightmare
Fix a diffeomorphism f of skew type or flow type. For α > 0,
denote
Γα (f ) = {x ∈ M : lim inf
1
log |Df n |E c (x) | > α.}
n
Proposition (Viana, Y)
There is δ such that
x ∈ Γα (f ) and every neighborhood
P for every
U of x, lim inf n1
length(f i (U)) > δ.
Theorem (Viana, Y)
There is N(α) such that Γα (f ) intersects each center leaf in at
most N points.
Jiagang Yang
Dynamical behavior of ergodic measure along the weak directio
Introduction
dynamical behavior along the center direction
Main results
Proof of Main results:
Example for weak absolute continuiation
Consider the Kan’s example, f0 : S 1 × [0, 1] → S 1 × [0, 1]:
t
)(1 − t)).
f0 (x, t) = (3x, t + cos(2πx)( 32
Let f be a C2 small perturbation of f which preserves S 1 × {0, 1}
and the fixed vertical line 0 × [0, 1], and ∂x f (0, 0) 6= ∂x f (0, 1).
Lemma (Viana, Y)
The disintegration of the volume along the center foliation of f is
absolutely continuous respect to the Lebesgue measure on the leaf,
but not equivalent to the Lebesgue measure. (i.e.) for any A in a
center leaf, we have
Lebc (A) = 0 ⇒ mc (A) = 0;
mc (A) = 0 ; Lebc (A) = 0.
Jiagang Yang
Dynamical behavior of ergodic measure along the weak directio
Introduction
dynamical behavior along the center direction
Main results
Proof of Main results:
Derived from Anosov diffeomorphisms
Let A ∈ Diff 1 (T 3 ) be a linear Anosov diffeomorphism with
exponents λ1 > λ2 > 0 > λ3 , f be a partially hyperbolic
diffeomorphism in the same isotopic class. πf the Franks
semi-conjugation between f and A.
Proposition
• f is dynamically coherent.
• πf maps center leaves to center leaves.
• there is Kf > 0 such that πf−1 (x) is a center segment with
length bounded by Kf .
Jiagang Yang
Dynamical behavior of ergodic measure along the weak directio
Introduction
dynamical behavior along the center direction
Main results
Proof of Main results:
Derived Anosov diffeomorphism
Proposition
If f is volume preserving with λc (f ) > λ2 , then F c has no
absolutely continuis property.
Example (Varão)
There is a volume preserving f (Anosov) such that the
disintegration of the volume along F c is singular but not atomic.
Example (Ponce, Tahzibi and Varão)
There is a volume preserving diffeomorphism f with metric entropy
λ1 such that, the disintegration of volume along F c is atomic.
Jiagang Yang
Dynamical behavior of ergodic measure along the weak directio
Introduction
dynamical behavior along the center direction
Main results
Proof of Main results:
Main results
Theorem 1 (Y):
Let µ be any ergodic measure of f with hµ (f ) > λ1 , if Γ is a
subset with µ(Γ) = 1, then Γ intersects almost every center leaf in
uncountably many points.
Jiagang Yang
Dynamical behavior of ergodic measure along the weak directio
Introduction
dynamical behavior along the center direction
Main results
Proof of Main results:
Main results
Theorem 1 (Y):
Let µ be any ergodic measure of f with hµ (f ) > λ1 , if Γ is a
subset with µ(Γ) = 1, then Γ intersects almost every center leaf in
uncountably many points.
Theorem 2(Y):
The Franks semi-conjugation πf : T 3 → T 3 induces a bijective
map between the ergodic measures with metric entropy larger than
λ1 .
Jiagang Yang
Dynamical behavior of ergodic measure along the weak directio
Introduction
dynamical behavior along the center direction
Main results
Proof of Main results:
Step I: reduce to the linear case
Since (πf )∗ preserves metric entropy, we only need to show the
following lemma:
Lemma
Let µ be an ergodic measure of A with metric entropy larger than
λ2 , then for any subset Γ with µ(Γ) = 1, it intersects almost every
center leaf in uncountable number of points.
Idea of Proof:
Consider a Markov partition for the center foliation of A. We will
show that the disintegration measures have positive dimension.
Jiagang Yang
Dynamical behavior of ergodic measure along the weak directio
Introduction
dynamical behavior along the center direction
Main results
Proof of Main results:
Dimensional theory
Recall the standard Ledrappier-Young’s dimensional theory
• h1 = λ1 γ1
• h2 = λ1 γ1 + λ2 γ̄2
• γ1 ≤ 1, γ̄2 ≤ 1.
where γ1 is the dimension of the disintegration along the strong
unstable direction, and γ2 is the dimension of the projection
measure on the center unstable direciton.
Application 1:
If hµ > λ2 , then γ1 > 0. Hence, every full µ measure subset
intersects the typical strong unstable leaf in a uncountable number
of points.
Jiagang Yang
Dynamical behavior of ergodic measure along the weak directio
Introduction
dynamical behavior along the center direction
Main results
Proof of Main results:
Dimensional theory along the center direction
We need note that γ̄2 cannot provide the information along the
center direction, since it is only about a projection measure.
Example
See the Shub, Wilkinson’s example: f is partially hyperbolic,
volume preserving with 1-D center, and λc > 0.
By the Pesin Formula, γ1 + γ̄2 = 2, hence γ1 = γ̄2 = 1
But every center leaf only contains finitely many generic points.
So we need to study γ2 .
Jiagang Yang
Dynamical behavior of ergodic measure along the weak directio
Introduction
dynamical behavior along the center direction
Main results
Proof of Main results:
Dimensional theory along the center direction
Lemma
λ1 ≥ λ1 γ̄1 ≥ hµ − hµc .
Finally, if there is a µ full measure subset which intersects every
center leaf in countably many points, then hµc = 0. So hµ ≤ λ1 .
Jiagang Yang
Dynamical behavior of ergodic measure along the weak directio
Introduction
dynamical behavior along the center direction
Main results
Proof of Main results:
Merci, mes amis!
Jiagang Yang
Dynamical behavior of ergodic measure along the weak directio